A new paper of mine (PRA 93, 012107 (2016), arXiv:1507.04083) just came out. The main theorem of the paper is not deep, but I think it’s a clarifying result within a formalism that is deep: ideal quantum Brownian motion (QBM) in symplectic generality. In this blog post, I’ll refresh you on ideal QBM, quote my abstract, explain the main result, and then — going beyond the paper — show how it’s related to the Kolmogorov-Sinai entropy and the speed at which macroscopic wavefunctions branch.
If you Google around for “quantum Brownian motion”, you’ll come across a bunch of definitions that have quirky features, and aren’t obviously related to each other. This is a shame. As I explained in an earlier blog post, ideal QBM is the generalization of the harmonic oscillator to open quantum systems. If you think harmonic oscillator are important, and you think decoherence is important, then you should understand ideal QBM.
Harmonic oscillators are ubiquitous in the world because all smooth potentials look quadratic locally. Exhaustively understanding harmonic oscillators is very valuable because they are exactly solvable in addition to being ubiquitous. In an almost identical way, all quantum Markovian degrees of freedom look locally like ideal QBM, and their completely positive (CP) dynamics can be solved exactly.
To get true generality, both harmonic oscillators and ideal QBM should be expressed in manifestly symplectic covariant form. Just like for Lorentz covariance, a dynamical equation that exhibits manifest symplectic covariance takes the same form under linear symplectic transformations on phase space. At a microscopic level, all physics is symplectic covariant (and Lorentz covariant), so this better hold.… [continue reading]